, , , .
:
= c1x1 + c2x2 +... + cnxn → max; | |||
| (2.4) | ||
xj ≥ 0, |
, (2.4), :
= b1y1 + b2y2 +... + bmym → min; | |||
| (2.5) | ||
yi ≥ 0, . |
.
1. , - .
2. .
3. - ≤, , , - ≥.
4. , .
5. .
6. .
.
1. , , :
. | (2.6) |
, .
2 ( ). , , (2.5) (2.6) , :
(2.7) | |
, , , .
. bi :
(2.8) |
. .
- . , .
|
|
.
() :
= 2x1 + 4x2 → max; | |||
| |||
x1 ≥ 0, x2 ≥ 0. |
.
= 120y1 + 72y2 + 10y3 → min; | |||
| |||
y1 ≥ 0, y2 ≥ 0, y3 ≥ 0. |
:
= (24, 4); = (1/3, 1/3, 0).
, 64.
.
1. . . , i- . , , .
( = 0), , .., , . , ( ) ( ) , ( = = 1/3).
, :
4ּ24 + 6ּ4 = 120, 2ּ24 + 6ּ4 = 72, 4 < 10. |
, , , .
2. . - , , . . - .
, . .
() () $1/3. , , 12 - $4 (1/3ּ12).
3. . , . , , - , . (Δj) :
(2.9) |
, Δj ≤ 0, , Δj > 0 .
|
|
. : . 3 , 4 1 . , , $3. ?
Δj (2.9):
Δj = 3ּ + 4ּ + 1ּ - 3 = 3ּ1/3 + 4ּ1/3 + 1ּ0 - 3 = -2/3,
Δj < 0, .
4. . , / , k- i- , ; , k- i- , , .
. , , , 1 - 1 -, .
, , ( , (2.6) (2.7)) , , .
.
1, 2, 3 4 I, II III. , , 1000, 600 150 .
( ) 2.12.
2.12 -
1 | 2 | 3 | 4 | ||
I | |||||
II | |||||
III | |||||
2,5 |
, , , .
.
= 6x1 + 2x2 + 2,5x3 + 4x4 → max; | |||
| |||
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0. |
( ):
= (0, 225, 0, 150); = 1050.
, .
= 1000y1 + 600y2 + 150y3 → min; | |||
| |||
y1 ≥ 0, y2 ≥ 0, y3 ≥ 0. |
, , :
5ּ0 + 225 + 2ּ150 < 1000, 4ּ0 + 2ּ225 + 2ּ0 + 150 = 600, 0 + 2ּ0 + 150 = 150. |
, , : = 0.
. . > 0 > 0, ( (2.7)). , = 0, :
2y2 = 2, y2 + y3 = 4. |
, : = 1, = 3.
|
|
:
= (0, 1, 3); = 1050.